On the deformation of a shear thinning viscoelastic drop in a steady electric field

Abstract

The deformation of viscoelastic drops under electric fields plays a crucial role in applications such as microfluidics, inkjet printing, and electrohydrodynamic manipulation of complex fluids. This study examines the deformation and breakup dynamics of a linear Phan-Thien-Tanner (LPTT) drop subjected to a uniform electric field using numerical simulations performed with the open-source solver Basilisk. Representative combinations of conductivity ratio (${\sigma }_r$) and permittivity ratio (${\epsilon }_r$) are chosen from six characteristic regions of the (${\sigma }_r$, ${\epsilon }_r$) phase space, $P{R}_A^{+}$, $P{R}_B^{+}$, $P{R}_A^{-}$, $P{R}_B^{-}$, $O{B}^{+}$, and $O{B}^{-}$. In regions where the first- and second-order deformation coefficients have the same sign ($P{R}_A^{-}$, $P{R}_B^{-}$, $O{B}^{+}$), the LPTT drops exhibit deformation dynamics that negligibley deviate from the Newtonian behavior. In the $P{R}_A^{+}$ region, drops deform into prolate spheroidal shapes below a critical electric capillary number and transition to stable multi-lobed shapes or breakup beyond this threshold. Increasing elasticity of drop opposes the deformation, thereby reducing deformation and increasing critical $C{a}_E$ with the Deborah number ($De$). In the $P{R}_B^{+}$ region, drops form prolate shapes below critical $C{a}_E$ and develop conical ends above it. The steady-state deformation exhibits a non-monotonic dependence on $De$, increasing at low $De$ and decreasing at higher values. A similar non-monotonic variation is also observed in critical $C{a}_E$. In the $O{B}^{-}$ region, LPTT drops attain oblate shapes below critical $C{a}_E$ and undergo breakup beyond it. The deformation magnitude shows a non-monotonic variation with $De$, increasing initially and decreasing at higher elasticity.

Prepared for the submission to Journal of Non-Newtonian Fluid Mechanics